Problem 4 - Advanced Engineering Mathematics

Question

Answer Problems \(1-8\) without referring back to the text. Fill in the blank or answer true/false. If \(f_{1}\) and \(f_{2}\) are linearly independent functions on an interval \(I\), then their Wronskian \(W\left(f_{1}, f_{2}\right) \neq 0\) for all \(x\) in \(I\).

Step-by-Step Solution

Answer

True. The Wronskian of linearly independent functions is not zero for any \(x\) in \(I\).

Step 1: Define Linear Independence

Two functions \(f_1\) and \(f_2\) are linearly independent on an interval \(I\) if there are no constants \(c_1\) and \(c_2\), not both zero, such that \(c_1f_1(x) + c_2f_2(x) = 0\) holds for all \(x\) in \(I\).

Step 2: Understand the Wronskian Concept

The Wronskian of two functions \(f_1\) and \(f_2\) is a determinant given by \(W(f_1, f_2) = f_1(x)f_2'(x) - f_2(x)f_1'(x)\). It is used to determine if functions are linearly independent.

Step 3: Relate Wronskian to Linear Independence

If two functions \(f_1\) and \(f_2\) are linearly independent on the interval \(I\), the Wronskian \(W(f_1, f_2)\) must be non-zero on the interval \(I\). This means \(W(f_1, f_2)eq 0\) for every \(x\) in \(I\).

Step 4: Evaluate the Given Statement

The statement "If \(f_{1}\) and \(f_{2}\) are linearly independent functions on an interval \(I\), then their Wronskian \(W\left(f_{1}, f_{2}\right) eq 0\) for all \(x\) in \(I\)" is truthfully expressing the relationship between Wronskian and linear independence.

Key Concepts

Wronskian

The Wronskian is an essential tool in calculus used to determine if a set of functions are linearly independent. For two functions, say \( f_1 \) and \( f_2 \), their Wronskian \( W(f_1, f_2) \) is calculated as follows:

• Take the derivative of both functions: \( f_1'(x) \) and \( f_2'(x) \).
• Form the "Wronskian determinant": \( W(f_1, f_2) = f_1(x)f_2'(x) - f_2(x)f_1'(x) \).

If \( W(f_1, f_2) \) evaluates to zero at every point in the interval \( I \), the functions are dependent, which means one can be expressed as a multiple of the other.
However, if \( W(f_1, f_2) \) is not always zero, it suggests the functions are linearly independent, meaning no such dependency exists. This property is notably helpful, especially in differential equations and mathematical modeling.

Determinants in Mathematics

Determinants are one of the fundamental concepts in linear algebra and play a significant role in various mathematical computations, including the Wronskian. The simplest way to explain a determinant is through a 2x2 matrix with elements \( a, b, c, \) and \( d \):

• The determinant of this matrix is calculated as \( ad - bc \).

Determinants are instrumental in determining matrix inversibility, solving systems of linear equations, and exploring vector spaces.
They serve more complex roles too. For example, in the context of functions, determinants help assess the linear (in)dependence of functions, thanks to their use in defining constructs like the Wronskian. They allow us to encapsulate the notion of volume, orientation, and can even define whether vectors span a volume in a multidimensional space.

Linearly Independent Functions

Understanding linearly independent functions is pivotal when working with systems of equations or modeling real-world scenarios with functions. Essentially, two or more functions are linearly independent if no function in the set can be written as a linear combination of the others. In mathematical terms, for functions \( f_1, f_2, \ldots, f_n \), they are linearly independent on interval \( I \) if:

• No constants \( c_1, c_2, \ldots, c_n \), not all zero, satisfy \( c_1f_1(x) + c_2f_2(x) + \ldots + c_nf_n(x) = 0 \) for all x in \( I \).

This concept is vital because linearly independent functions form a basis for function spaces. This means that any other function in that space can be expressed as a combination of these independent functions.
In practice, establishing the independence of functions often involves checking the Wronskian, as a non-zero result implies independence. This method becomes invaluable when working with solutions to differential equations, where identifying independent solutions can be crucial.